Shrink Wrap Modifier and Distortion
I went through the options, with the same successs :( so ... what about Booleans?


to help a bit topology, use Simple Deform Modifier > Twist
ShrinkWrap > 
Smooth Modifier > Repeat 5
add a Cube and for Cylinder add Boolean Modifier > Intersect


(if you plan to move Cylinder, would be better to parent (Ctrl+P) Cube to Cylinder)




to smooth surface - search for Smooth Shading
fix for edges shading - go to Properties Editor > Object Data > Normals > Auto Smooth


(like that Subsurface Modifier probably is not needed in this case)